Detailed proof: (a) If at least one xi is zero, then the left-hand side of the Ky Fan inequality is zero and the inequality is proved. Equality holds if and only if the right-hand side is also zero, which is the case when γixi = 0 for all i = 1, . . ., n.

(b) Assume now that all xi > 0. If there is an i with γi = 0, then the corresponding xi > 0 has no effect on either side of the inequality, hence the ith term can be omitted. Therefore, we may assume that γi > 0 for all i in the following. If x1 = x2 = . . . = xn, then equality holds. It remains to show strict inequality if not all xi are equal.

The function f is strictly concave on (0,½], because we have for its second derivative

A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications". This second inequality is equivalent to the Brouwer Fixed Point Theorem, but is often more convenient. Let S be a compactconvex subset of a finite-dimensional vector spaceV, and let f(x,y) be a function from S × S to the real numbers that is lower semicontinuous in x, concave in y and has f(z,z) ≤ 0 for all z in S. Then there exists x* ∈ S such that for all y ∈ S, f( x* , y ) ≤ 0 . This Ky Fan Inequality is used to establish the existence of equilibria in various games studied in economics.